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exametrika
Overview
The exametrika package provides comprehensive Test Data
Engineering tools for analyzing educational test data. Based on the
methods described in Shojima (2022), this package enables researchers
and practitioners to:
- Analyze test response patterns and item characteristics
- Classify respondents using various psychometric models
- Investigate latent structures in test data
- Examine local dependencies between items
- Perform network analysis of item relationships
The package implements both traditional psychometric approaches and advanced statistical methods, making it suitable for various assessment and research purposes.
Features
The package implements various psychometric models and techniques:
Classical Methods
- Classical Test Theory (CTT)
- Item difficulty and discrimination
- Test reliability and validity
- Item Response Theory (IRT)
- 2PL, 3PL, and 4PL models
- Item characteristic curves
- Test information functions
Latent Structure Analysis
- Latent Class Analysis (LCA)
- Class membership estimation
- Item response profiles
- Latent Rank Analysis (LRA)
- Ordered latent classes
- Rank transition probabilities
- Biclustering and Ranklustering
- Simultaneous clustering of items and examinees
- Field-specific response patterns
- Infinite Relational Model (IRM)
- Optimal class/field determination
- Nonparametric clustering
Advanced Network Models
- Bayesian Network Analysis
- Structure Learning
- Genetic Algorithm approach
- Population-Based Incremental Learning (PBIL)
- Conditional probability estimation
- Structure Learning
- Local Dependence Models
- Local Dependence Latent Rank Analysis (LDLRA)
- Local Dependence Biclustering (LDB)
- Bicluster Network Model (BINET)
Model Overview
Local Dependence Models
The package implements three complementary approaches to modeling local dependencies in test data:
- LDLRA (Local Dependence Latent Rank Analysis)
- Analyzes how item dependencies change across different proficiency ranks
- Suitable when item relationships are expected to vary by student ability level
- Combines the strengths of LRA and Bayesian Networks
- LDB (Local Dependence Biclustering)
- Focuses on relationships between item fields within each rank
- Optimal when items naturally form groups (fields) with hierarchical relationships
- Integrates biclustering with field-level dependency structures
- BINET (Bicluster Network Model)
- Examines class transitions within each field
- Best for understanding complex patterns of class progression
- Combines biclustering with class-level network analysis
Background
Exametrika was originally developed and published as a Mathematica and Excel Add-in. For additional information about Exametrika, visit:
Installation
The development version of Exametrika can be installed from GitHub:
# Install devtools if not already installed
if (!require("devtools")) install.packages("devtools")
# Install Exametrika
devtools::install_github("kosugitti/exametrika")Dependencies
The package requires:
- R (>= 4.1.0)
- igraph (for network analysis)
- Other dependencies are automatically installed
Data Format and Usage
Basic Usage
library(exametrika)Data Requirements
Exametrika accepts both binary and polytomous response data:
- Binary data (0/1)
- 0: Incorrect answer
- 1: Correct answer
- Polytomous data
- Ordinal response categories
- Multiple score levels
- Missing values
- NA values supported
- Custom missing value codes can be specified
Input Data Specifications
The package accepts data in several formats with the following features:
- Data Structure
- Matrix or data.frame format
- Response data (binary or polytomous)
- Flexible handling of missing values
- Support for various data types and structures
- Optional Components
- Examinee ID column (default: first column)
- Item weights (default: all weights = 1)
- Item labels (default: sequential numbers)
- Missing value indicator matrix
Note: Some analysis methods may have specific data type requirements. Please refer to each function’s documentation for detailed requirements.
Data Formatting
The dataFormat function preprocesses input data for
analysis:
- Functions
- Extracts and validates ID vectors if present
- Processes item labels or assigns sequential numbers
- Creates response data matrix U
- Generates missing value indicator matrix Z
- Handles item weights
- Converts data to appropriate format for analysis
Example:
# Format raw data for analysis
data <- dataFormat(J15S500) # Using sample dataset
str(data) # View structure of formatted data## List of 7
## $ ID : chr [1:500] "Student001" "Student002" "Student003" "Student004" ...
## $ ItemLabel : chr [1:15] "Item01" "Item02" "Item03" "Item04" ...
## $ Z : num [1:500, 1:15] 1 1 1 1 1 1 1 1 1 1 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : NULL
## .. ..$ : chr [1:15] "Item01" "Item02" "Item03" "Item04" ...
## $ w : num [1:15] 1 1 1 1 1 1 1 1 1 1 ...
## $ response.type: chr "binary"
## $ categories : Named int [1:15] 2 2 2 2 2 2 2 2 2 2 ...
## ..- attr(*, "names")= chr [1:15] "Item01" "Item02" "Item03" "Item04" ...
## $ U : num [1:500, 1:15] 0 1 1 1 1 1 0 0 1 1 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : NULL
## .. ..$ : chr [1:15] "Item01" "Item02" "Item03" "Item04" ...
## - attr(*, "class")= chr [1:2] "exametrika" "exametrikaData"Sample Datasets
The package includes various sample datasets from Shojima (2022) for testing and learning:
- Naming Convention: JxxSxxx format
- J: Number of items (e.g., J15 = 15 items)
- S: Sample size (e.g., S500 = 500 examinees)
Available datasets:
- J5S10: Very small dataset (5 items, 10 examinees)
- Useful for quick testing and understanding basic concepts
- J12S5000: Large sample dataset (12 items, 5000 examinees)
- Suitable for LDLRA and other advanced analyses
- J14S500: Medium dataset (14 items, 500 examinees)
- J15S500: Medium dataset (15 items, 500 examinees)
- Often used in IRT and LCA examples
- J20S400: Medium dataset (20 items, 400 examinees)
- J35S515: Large item dataset (35 items, 515 examinees)
- Used in Biclustering and network model examples
Examples
Test Statistics
TestStatistics(J15S500)## Test Statics
## value
## TestLength 15.0000000
## SampleSize 500.0000000
## Mean 9.6640000
## SEofMean 0.1190738
## Variance 7.0892826
## SD 2.6625707
## Skewness -0.4116220
## Kurtosis -0.4471624
## Min 2.0000000
## Max 15.0000000
## Range 13.0000000
## Q1.25% 8.0000000
## Median.50% 10.0000000
## Q3.75% 12.0000000
## IQR.75% 4.0000000
## Stanine.4% 5.0000000
## Stanine.11% 6.0000000
## Stanine.23% 7.0000000
## Stanine.40% 9.0000000
## Stanine.60% 11.0000000
## Stanine.77% 12.0000000
## Stanine.89% 13.0000000
## Stanine.96% 14.0000000ItemStatistics
ItemStatistics(J15S500)## Item Statics
## ItemLabel NR CRR ODDs Threshold Entropy ITCrr
## 1 Item01 500 0.746 2.937 -0.662 0.818 0.375
## 2 Item02 500 0.754 3.065 -0.687 0.805 0.393
## 3 Item03 500 0.726 2.650 -0.601 0.847 0.321
## 4 Item04 500 0.776 3.464 -0.759 0.767 0.503
## 5 Item05 500 0.804 4.102 -0.856 0.714 0.329
## 6 Item06 500 0.864 6.353 -1.098 0.574 0.377
## 7 Item07 500 0.716 2.521 -0.571 0.861 0.483
## 8 Item08 500 0.588 1.427 -0.222 0.978 0.405
## 9 Item09 500 0.364 0.572 0.348 0.946 0.225
## 10 Item10 500 0.662 1.959 -0.418 0.923 0.314
## 11 Item11 500 0.286 0.401 0.565 0.863 0.455
## 12 Item12 500 0.274 0.377 0.601 0.847 0.468
## 13 Item13 500 0.634 1.732 -0.342 0.948 0.471
## 14 Item14 500 0.764 3.237 -0.719 0.788 0.485
## 15 Item15 500 0.706 2.401 -0.542 0.874 0.413CTT
CTT(J15S500)## Realiability
## name value
## 1 Alpha(Covariance) 0.625
## 2 Alpha(Phi) 0.630
## 3 Alpha(Tetrachoric) 0.771
## 4 Omega(Covariance) 0.632
## 5 Omega(Phi) 0.637
## 6 Omega(Tetrachoric) 0.779
##
## Reliability Excluding Item
## IfDeleted Alpha.Covariance Alpha.Phi Alpha.Tetrachoric
## 1 Item01 0.613 0.618 0.762
## 2 Item02 0.609 0.615 0.759
## 3 Item03 0.622 0.628 0.770
## 4 Item04 0.590 0.595 0.742
## 5 Item05 0.617 0.624 0.766
## 6 Item06 0.608 0.613 0.754
## 7 Item07 0.594 0.600 0.748
## 8 Item08 0.611 0.616 0.762
## 9 Item09 0.642 0.645 0.785
## 10 Item10 0.626 0.630 0.773
## 11 Item11 0.599 0.606 0.751
## 12 Item12 0.597 0.603 0.748
## 13 Item13 0.597 0.604 0.753
## 14 Item14 0.593 0.598 0.745
## 15 Item15 0.607 0.612 0.759IRT
The IRT function estimates the number of parameters using a logistic
model, which can be specified using the model option. It
supports 2PL, 3PL, and 4PL models.
result.IRT <- IRT(J15S500, model = 3)
result.IRT## Item Parameters
## slope location lowerAsym PSD(slope) PSD(location) PSD(lowerAsym)
## Item01 0.818 -0.834 0.2804 0.182 0.628 0.1702
## Item02 0.860 -1.119 0.1852 0.157 0.471 0.1488
## Item03 0.657 -0.699 0.3048 0.162 0.798 0.1728
## Item04 1.550 -0.949 0.1442 0.227 0.216 0.1044
## Item05 0.721 -1.558 0.2584 0.148 0.700 0.1860
## Item06 1.022 -1.876 0.1827 0.171 0.423 0.1577
## Item07 1.255 -0.655 0.1793 0.214 0.289 0.1165
## Item08 0.748 -0.155 0.1308 0.148 0.394 0.1077
## Item09 1.178 2.287 0.2930 0.493 0.423 0.0440
## Item10 0.546 -0.505 0.2221 0.131 0.779 0.1562
## Item11 1.477 1.090 0.0628 0.263 0.120 0.0321
## Item12 1.479 1.085 0.0462 0.245 0.115 0.0276
## Item13 0.898 -0.502 0.0960 0.142 0.272 0.0858
## Item14 1.418 -0.788 0.2260 0.248 0.291 0.1252
## Item15 0.908 -0.812 0.1531 0.159 0.383 0.1254
##
## Item Fit Indices
## model_log_like bench_log_like null_log_like model_Chi_sq null_Chi_sq model_df
## Item01 -262.979 -240.190 -283.343 45.578 86.307 11
## Item02 -253.405 -235.436 -278.949 35.937 87.025 11
## Item03 -280.640 -260.906 -293.598 39.468 65.383 11
## Item04 -204.884 -192.072 -265.962 25.623 147.780 11
## Item05 -232.135 -206.537 -247.403 51.196 81.732 11
## Item06 -173.669 -153.940 -198.817 39.459 89.755 11
## Item07 -250.905 -228.379 -298.345 45.053 139.933 11
## Item08 -314.781 -293.225 -338.789 43.111 91.127 11
## Item09 -321.920 -300.492 -327.842 42.856 54.700 11
## Item10 -309.318 -288.198 -319.850 42.240 63.303 11
## Item11 -248.409 -224.085 -299.265 48.647 150.360 11
## Item12 -238.877 -214.797 -293.598 48.161 157.603 11
## Item13 -293.472 -262.031 -328.396 62.882 132.730 11
## Item14 -223.473 -204.953 -273.212 37.040 136.519 11
## Item15 -271.903 -254.764 -302.847 34.279 96.166 11
## null_df NFI RFI IFI TLI CFI RMSEA AIC CAIC BIC
## Item01 13 0.472 0.376 0.541 0.443 0.528 0.079 23.578 -22.805 -22.783
## Item02 13 0.587 0.512 0.672 0.602 0.663 0.067 13.937 -32.446 -32.424
## Item03 13 0.396 0.287 0.477 0.358 0.457 0.072 17.468 -28.915 -28.893
## Item04 13 0.827 0.795 0.893 0.872 0.892 0.052 3.623 -42.759 -42.737
## Item05 13 0.374 0.260 0.432 0.309 0.415 0.086 29.196 -17.186 -17.164
## Item06 13 0.560 0.480 0.639 0.562 0.629 0.072 17.459 -28.924 -28.902
## Item07 13 0.678 0.620 0.736 0.683 0.732 0.079 23.053 -23.330 -23.308
## Item08 13 0.527 0.441 0.599 0.514 0.589 0.076 21.111 -25.272 -25.250
## Item09 13 0.217 0.074 0.271 0.097 0.236 0.076 20.856 -25.527 -25.505
## Item10 13 0.333 0.211 0.403 0.266 0.379 0.075 20.240 -26.143 -26.121
## Item11 13 0.676 0.618 0.730 0.676 0.726 0.083 26.647 -19.736 -19.714
## Item12 13 0.694 0.639 0.747 0.696 0.743 0.082 26.161 -20.222 -20.200
## Item13 13 0.526 0.440 0.574 0.488 0.567 0.097 40.882 -5.501 -5.479
## Item14 13 0.729 0.679 0.793 0.751 0.789 0.069 15.040 -31.343 -31.321
## Item15 13 0.644 0.579 0.727 0.669 0.720 0.065 12.279 -34.104 -34.082
##
## Model Fit Indices
## value
## model_log_like -3880.769
## bench_log_like -3560.005
## null_log_like -4350.217
## model_Chi_sq 641.528
## null_Chi_sq 1580.424
## model_df 165.000
## null_df 195.000
## NFI 0.594
## RFI 0.520
## IFI 0.663
## TLI 0.594
## CFI 0.656
## RMSEA 0.076
## AIC 311.528
## CAIC -384.212
## BIC -383.882The estimated population of subjects is included in the returned object.
head(result.IRT$ability)## tmp$ID EAP PSD
## 1 Student001 -0.75526283 0.5805695
## 2 Student002 -0.17398774 0.5473605
## 3 Student003 0.01382272 0.5530503
## 4 Student004 0.57627973 0.5749113
## 5 Student005 -0.97449585 0.5915604
## 6 Student006 0.85233108 0.5820541The plots offer options for Item Characteristic Curves (ICC), Item
Information Curves (IIC), and Test Information Curves (TIC), which can
be specified through options. Items can be specified using the
items argument, and if not specified, plots will be drawn
for all items. The number of rows and columns for dividing the plotting
area can be specified using nr and nc,
respectively.
plot(result.IRT, type = "ICC", items = 1:6, nc = 2, nr = 3)
plot(result.IRT, type = "IIC", items = 1:6, nc = 2, nr = 3)
plot(result.IRT, type = "TIC")
LCA Example
Latent Class Analysis requires specifying the dataset and the number of classes.
LCA(J15S500, ncls = 5)##
## Item Reference Profile
## IRP1 IRP2 IRP3 IRP4 IRP5
## Item01 0.5185 0.6996 0.76358 0.856 0.860
## Item02 0.5529 0.6276 0.81161 0.888 0.855
## Item03 0.7959 0.3205 0.93735 0.706 0.849
## Item04 0.5069 0.5814 0.86940 0.873 1.000
## Item05 0.6154 0.7523 0.94673 0.789 0.886
## Item06 0.6840 0.7501 0.94822 1.000 0.907
## Item07 0.4832 0.4395 0.83377 0.874 0.900
## Item08 0.3767 0.3982 0.62563 0.912 0.590
## Item09 0.3107 0.3980 0.26616 0.165 0.673
## Item10 0.5290 0.5341 0.76134 0.677 0.781
## Item11 0.1007 0.0497 0.00132 0.621 0.623
## Item12 0.0355 0.1673 0.15911 0.296 0.673
## Item13 0.2048 0.5490 0.89445 0.672 0.784
## Item14 0.3508 0.7384 0.77159 0.904 1.000
## Item15 0.3883 0.6077 0.82517 0.838 0.823
##
## Test Profile
## Class 1 Class 2 Class 3 Class 4 Class 5
## Test Reference Profile 6.453 7.613 10.415 11.072 12.205
## Latent Class Ditribution 87.000 97.000 125.000 91.000 100.000
## Class Membership Distribution 90.372 97.105 105.238 102.800 104.484
##
## Item Fit Indices
## model_log_like bench_log_like null_log_like model_Chi_sq null_Chi_sq model_df
## Item01 -264.179 -240.190 -283.343 47.978 86.307 9
## Item02 -256.363 -235.436 -278.949 41.853 87.025 9
## Item03 -237.888 -260.906 -293.598 -46.037 65.383 9
## Item04 -208.536 -192.072 -265.962 32.928 147.780 9
## Item05 -226.447 -206.537 -247.403 39.819 81.732 9
## Item06 -164.762 -153.940 -198.817 21.644 89.755 9
## Item07 -249.377 -228.379 -298.345 41.997 139.933 9
## Item08 -295.967 -293.225 -338.789 5.483 91.127 9
## Item09 -294.250 -300.492 -327.842 -12.484 54.700 9
## Item10 -306.985 -288.198 -319.850 37.574 63.303 9
## Item11 -187.202 -224.085 -299.265 -73.767 150.360 9
## Item12 -232.307 -214.797 -293.598 35.020 157.603 9
## Item13 -267.647 -262.031 -328.396 11.232 132.730 9
## Item14 -203.468 -204.953 -273.212 -2.969 136.519 9
## Item15 -268.616 -254.764 -302.847 27.705 96.166 9
## null_df NFI RFI IFI TLI CFI RMSEA AIC CAIC BIC
## Item01 13 0.444 0.197 0.496 0.232 0.468 0.093 29.978 -7.972 -7.954
## Item02 13 0.519 0.305 0.579 0.359 0.556 0.086 23.853 -14.097 -14.079
## Item03 13 1.000 1.000 1.000 1.000 1.000 0.000 -64.037 -101.987 -101.969
## Item04 13 0.777 0.678 0.828 0.744 0.822 0.073 14.928 -23.022 -23.004
## Item05 13 0.513 0.296 0.576 0.352 0.552 0.083 21.819 -16.130 -16.112
## Item06 13 0.759 0.652 0.843 0.762 0.835 0.053 3.644 -34.305 -34.287
## Item07 13 0.700 0.566 0.748 0.625 0.740 0.086 23.997 -13.952 -13.934
## Item08 13 0.940 0.913 1.000 1.000 1.000 0.000 -12.517 -50.466 -50.448
## Item09 13 1.000 1.000 1.000 1.000 1.000 0.000 -30.484 -68.433 -68.415
## Item10 13 0.406 0.143 0.474 0.179 0.432 0.080 19.574 -18.375 -18.357
## Item11 13 1.000 1.000 1.000 1.000 1.000 0.000 -91.767 -129.716 -129.698
## Item12 13 0.778 0.679 0.825 0.740 0.820 0.076 17.020 -20.930 -20.912
## Item13 13 0.915 0.878 0.982 0.973 0.981 0.022 -6.768 -44.717 -44.699
## Item14 13 1.000 1.000 1.000 1.000 1.000 0.000 -20.969 -58.919 -58.901
## Item15 13 0.712 0.584 0.785 0.675 0.775 0.065 9.705 -28.244 -28.226
##
## Model Fit Indices
## Number of Latent class: 5
## Number of EM cycle: 73
## value
## model_log_like -3663.994
## bench_log_like -3560.005
## null_log_like -4350.217
## model_Chi_sq 207.977
## null_Chi_sq 1580.424
## model_df 135.000
## null_df 195.000
## NFI 0.868
## RFI 0.810
## IFI 0.950
## TLI 0.924
## CFI 0.947
## RMSEA 0.033
## AIC -62.023
## CAIC -631.265
## BIC -630.995The returned object contains the Class Membership Matrix, which indicates which latent class each subject belongs to. The Estimate includes the one with the highest membership probability.
result.LCA <- LCA(J15S500, ncls = 5)
head(result.LCA$Students)## Membership 1 Membership 2 Membership 3 Membership 4 Membership 5 Estimate
## [1,] 0.7839477684 0.171152798 0.004141844 4.075759e-02 3.744590e-12 1
## [2,] 0.0347378747 0.051502214 0.836022799 7.773694e-02 1.698776e-07 3
## [3,] 0.0146307878 0.105488644 0.801853496 3.343026e-02 4.459682e-02 3
## [4,] 0.0017251650 0.023436459 0.329648386 3.656488e-01 2.795412e-01 4
## [5,] 0.2133830569 0.784162066 0.001484616 2.492073e-08 9.702355e-04 2
## [6,] 0.0003846482 0.001141448 0.001288901 8.733869e-01 1.237981e-01 4The plots offer options for IRP, CMP, TRP, and LCD. For more details on each, please refer to Shojima (2022).
plot(result.LCA, type = "IRP", items = 1:6, nc = 2, nr = 3)
plot(result.LCA, type = "CMP", students = 1:9, nc = 3, nr = 3)
plot(result.LCA, type = "TRP")
plot(result.LCA, type = "LCD")
LRA Example
Latent Rank Analysis requires specifying the dataset and the number of classes.
LRA(J15S500, nrank = 6)## estimating method is GTMItem Reference Profile
## IRP1 IRP2 IRP3 IRP4 IRP5 IRP6
## Item01 0.5851 0.6319 0.708 0.787 0.853 0.898
## Item02 0.5247 0.6290 0.755 0.845 0.883 0.875
## Item03 0.6134 0.6095 0.708 0.773 0.801 0.839
## Item04 0.4406 0.6073 0.794 0.882 0.939 0.976
## Item05 0.6465 0.7452 0.821 0.837 0.862 0.905
## Item06 0.6471 0.7748 0.911 0.967 0.963 0.915
## Item07 0.4090 0.5177 0.720 0.840 0.890 0.900
## Item08 0.3375 0.4292 0.602 0.713 0.735 0.698
## Item09 0.3523 0.3199 0.298 0.282 0.377 0.542
## Item10 0.4996 0.5793 0.686 0.729 0.717 0.753
## Item11 0.0958 0.0793 0.136 0.286 0.472 0.617
## Item12 0.0648 0.0982 0.156 0.239 0.421 0.636
## Item13 0.2908 0.4842 0.715 0.773 0.750 0.778
## Item14 0.4835 0.5949 0.729 0.849 0.933 0.977
## Item15 0.3981 0.5745 0.756 0.827 0.835 0.834
##
## Item Reference Profile Indices
## Alpha A Beta B Gamma C
## Item01 3 0.0786 1 0.585 0.0 0.00000
## Item02 2 0.1264 1 0.525 0.2 -0.00787
## Item03 2 0.0987 2 0.610 0.2 -0.00391
## Item04 2 0.1864 1 0.441 0.0 0.00000
## Item05 1 0.0987 1 0.647 0.0 0.00000
## Item06 2 0.1362 1 0.647 0.4 -0.05198
## Item07 2 0.2028 2 0.518 0.0 0.00000
## Item08 2 0.1731 2 0.429 0.2 -0.03676
## Item09 5 0.1646 6 0.542 0.6 -0.07002
## Item10 2 0.1069 1 0.500 0.2 -0.01244
## Item11 4 0.1867 5 0.472 0.2 -0.01650
## Item12 5 0.2146 5 0.421 0.0 0.00000
## Item13 2 0.2310 2 0.484 0.2 -0.02341
## Item14 2 0.1336 1 0.484 0.0 0.00000
## Item15 2 0.1817 2 0.574 0.2 -0.00123
##
## Test Profile
## Class 1 Class 2 Class 3 Class 4 Class 5 Class 6
## Test Reference Profile 6.389 7.675 9.496 10.631 11.432 12.144
## Latent Class Ditribution 96.000 60.000 91.000 77.000 73.000 103.000
## Class Membership Distribution 83.755 78.691 81.853 84.918 84.238 86.545
##
## Item Fit Indices
## model_log_like bench_log_like null_log_like model_Chi_sq null_Chi_sq model_df
## Item01 -264.495 -240.190 -283.343 48.611 86.307 9.233
## Item02 -253.141 -235.436 -278.949 35.409 87.025 9.233
## Item03 -282.785 -260.906 -293.598 43.758 65.383 9.233
## Item04 -207.082 -192.072 -265.962 30.021 147.780 9.233
## Item05 -234.902 -206.537 -247.403 56.730 81.732 9.233
## Item06 -168.218 -153.940 -198.817 28.556 89.755 9.233
## Item07 -250.864 -228.379 -298.345 44.970 139.933 9.233
## Item08 -312.621 -293.225 -338.789 38.791 91.127 9.233
## Item09 -317.600 -300.492 -327.842 34.216 54.700 9.233
## Item10 -309.654 -288.198 -319.850 42.910 63.303 9.233
## Item11 -242.821 -224.085 -299.265 37.472 150.360 9.233
## Item12 -236.522 -214.797 -293.598 43.451 157.603 9.233
## Item13 -287.782 -262.031 -328.396 51.502 132.730 9.233
## Item14 -221.702 -204.953 -273.212 33.499 136.519 9.233
## Item15 -267.793 -254.764 -302.847 26.059 96.166 9.233
## null_df NFI RFI IFI TLI CFI RMSEA AIC CAIC BIC
## Item01 13 0.437 0.207 0.489 0.244 0.463 0.092 30.146 -8.785 -8.767
## Item02 13 0.593 0.427 0.664 0.502 0.646 0.075 16.944 -21.987 -21.969
## Item03 13 0.331 0.058 0.385 0.072 0.341 0.087 25.293 -13.638 -13.620
## Item04 13 0.797 0.714 0.850 0.783 0.846 0.067 11.555 -27.375 -27.357
## Item05 13 0.306 0.023 0.345 0.027 0.309 0.102 38.264 -0.667 -0.648
## Item06 13 0.682 0.552 0.760 0.646 0.748 0.065 10.091 -28.840 -28.822
## Item07 13 0.679 0.548 0.727 0.604 0.718 0.088 26.504 -12.427 -12.408
## Item08 13 0.574 0.401 0.639 0.467 0.622 0.080 20.326 -18.605 -18.587
## Item09 13 0.374 0.119 0.451 0.156 0.401 0.074 15.751 -23.180 -23.162
## Item10 13 0.322 0.046 0.377 0.057 0.330 0.085 24.445 -14.486 -14.467
## Item11 13 0.751 0.649 0.800 0.711 0.794 0.078 19.006 -19.925 -19.906
## Item12 13 0.724 0.612 0.769 0.667 0.763 0.086 24.985 -13.946 -13.927
## Item13 13 0.612 0.454 0.658 0.503 0.647 0.096 33.037 -5.894 -5.875
## Item14 13 0.755 0.654 0.809 0.723 0.804 0.073 15.034 -23.897 -23.879
## Item15 13 0.729 0.618 0.806 0.715 0.798 0.060 7.593 -31.338 -31.319
##
## Model Fit Indices
## Number of Latent class: 6
## Number of EM cycle: 17
## value
## model_log_like -3857.982
## bench_log_like -3560.005
## null_log_like -4350.217
## model_Chi_sq 595.954
## null_Chi_sq 1580.424
## model_df 138.491
## null_df 195.000
## NFI 0.623
## RFI 0.469
## IFI 0.683
## TLI 0.535
## CFI 0.670
## RMSEA 0.081
## AIC 318.973
## CAIC -264.989
## BIC -264.712The estimated subject rank membership probabilities and plots are almost the same as those in LCA (Latent Class Analysis). Since a ranking is assumed for the latent classes, rank-up odds and rank-down odds are calculated.
result.LRA <- LRA(J15S500, nrank = 6)
head(result.LRA$Students)## Membership 1 Membership 2 Membership 3 Membership 4 Membership 5
## Student001 0.2704649921 0.357479353 0.27632327 0.084988078 0.010069050
## Student002 0.0276546965 0.157616072 0.47438958 0.279914853 0.053715813
## Student003 0.0228189795 0.138860955 0.37884545 0.284817610 0.120794858
## Student004 0.0020140858 0.015608542 0.09629429 0.216973334 0.362406292
## Student005 0.5582996437 0.397431414 0.03841668 0.003365601 0.001443909
## Student006 0.0003866603 0.003168853 0.04801344 0.248329964 0.428747502
## Membership 6 Estimate Rank-Up Odds Rank-Down Odds
## Student001 0.0006752546 2 0.7729769 0.7565891
## Student002 0.0067089816 3 0.5900527 0.3322503
## Student003 0.0538621490 3 0.7518042 0.3665372
## Student004 0.3067034562 5 0.8462973 0.5987019
## Student005 0.0010427491 1 0.7118604 NA
## Student006 0.2713535842 5 0.6328983 0.5791986plot(result.LRA, type = "IRP", items = 1:6, nc = 2, nr = 3)
plot(result.LRA, type = "RMP", students = 1:9, nc = 3, nr = 3)
plot(result.LRA, type = "TRP")
plot(result.LRA, type = "LRD")
Biclustering/Ranklustering
Biclustering and Ranklustering algorithms are almost the same,
differing only in whether they include a filtering matrix or not. The
difference is specified using the method option in the
Biclustering() function. For more details, please refer to
the help documentation.
Biclustering(J35S515, nfld = 5, ncls = 6, method = "B")## Biclustering is chosen.## iter 1 logLik -7966.66iter 2 logLik -7442.38iter 3 logLik -7266.35iter 4 logLik -7151.01iter 5 logLik -7023.94iter 6 logLik -6984.82iter 7 logLik -6950.27iter 8 logLik -6939.34iter 9 logLik -6930.89iter 10 logLik -6923.5iter 11 logLik -6914.56iter 12 logLik -6908.89iter 13 logLik -6906.84iter 14 logLik -6905.39iter 15 logLik -6904.24iter 16 logLik -6903.28iter 17 logLik -6902.41iter 18 logLik -6901.58iter 19 logLik -6900.74iter 20 logLik -6899.86iter 21 logLik -6898.9iter 22 logLik -6897.84iter 23 logLik -6896.66iter 24 logLik -6895.35iter 25 logLik -6893.92iter 26 logLik -6892.4iter 27 logLik -6890.85iter 28 logLik -6889.32iter 29 logLik -6887.9iter 30 logLik -6886.66iter 31 logLik -6885.67iter 32 logLik -6884.98iter 33 logLik -6884.58iter 33 logLik -6884.58## Bicluster Matrix Profile
## Class1 Class2 Class3 Class4 Class5 Class6
## Field1 0.6236 0.8636 0.8718 0.898 0.952 1.000
## Field2 0.0627 0.3332 0.4255 0.919 0.990 1.000
## Field3 0.2008 0.5431 0.2281 0.475 0.706 1.000
## Field4 0.0495 0.2455 0.0782 0.233 0.648 0.983
## Field5 0.0225 0.0545 0.0284 0.043 0.160 0.983
##
## Field Reference Profile Indices
## Alpha A Beta B Gamma C
## Field1 1 0.240 1 0.624 0.0 0.0000
## Field2 3 0.493 3 0.426 0.0 0.0000
## Field3 1 0.342 4 0.475 0.2 -0.3149
## Field4 4 0.415 5 0.648 0.2 -0.1673
## Field5 5 0.823 5 0.160 0.2 -0.0261
##
## Class 1 Class 2 Class 3 Class 4 Class 5 Class 6
## Test Reference Profile 4.431 11.894 8.598 16.002 23.326 34.713
## Latent Class Ditribution 157.000 64.000 82.000 106.000 89.000 17.000
## Class Membership Distribution 146.105 73.232 85.753 106.414 86.529 16.968
## Latent Field Distribution
## Field 1 Field 2 Field 3 Field 4 Field 5
## N of Items 3 8 7 10 7
##
## Model Fit Indices
## Number of Latent Class : 6
## Number of Latent Field: 5
## Number of EM cycle: 33
## value
## model_log_like -6884.582
## bench_log_like -5891.314
## null_log_like -9862.114
## model_Chi_sq 1986.535
## null_Chi_sq 7941.601
## model_df 1160.000
## null_df 1155.000
## NFI 0.750
## RFI 0.751
## IFI 0.878
## TLI 0.879
## CFI 0.878
## RMSEA 0.037
## AIC -333.465
## CAIC -5258.949
## BIC -5256.699result.Ranklustering <- Biclustering(J35S515, nfld = 5, ncls = 6, method = "R")## Ranklustering is chosen.## iter 1 logLik -8097.56iter 2 logLik -7669.21iter 3 logLik -7586.72iter 4 logLik -7568.24iter 5 logLik -7561.02iter 6 logLik -7557.34iter 7 logLik -7557.36iter 7 logLik -7557.36
## Strongly ordinal alignment condition was satisfied.plot(result.Ranklustering, type = "Array")
plot(result.Ranklustering, type = "FRP", nc = 2, nr = 3)
plot(result.Ranklustering, type = "RMP", students = 1:9, nc = 3, nr = 3)
plot(result.Ranklustering, type = "LRD")
To find the optimal number of classes and the optimal number of fields, the Infinite Relational Model is available.
result.IRM <- IRM(J35S515, gamma_c = 1, gamma_f = 1, verbose = TRUE)## iter 1 Exact match count of field elements. 0 nfld 15 ncls 30## iter 2 Exact match count of field elements. 0 nfld 12 ncls 27## iter 3 Exact match count of field elements. 1 nfld 12 ncls 24## iter 4 Exact match count of field elements. 2 nfld 12 ncls 23## iter 5 Exact match count of field elements. 3 nfld 12 ncls 23## iter 6 Exact match count of field elements. 0 nfld 12 ncls 23## iter 7 Exact match count of field elements. 1 nfld 12 ncls 23## iter 8 Exact match count of field elements. 2 nfld 12 ncls 23## iter 9 Exact match count of field elements. 3 nfld 12 ncls 21## iter 10 Exact match count of field elements. 4 nfld 12 ncls 21## iter 11 Exact match count of field elements. 5 nfld 12 ncls 21## The minimum class member count is under the setting value.
## bic -99592.5 nclass 21## The minimum class member count is under the setting value.
## bic -99980.4 nclass 20## The minimum class member count is under the setting value.
## bic -99959.7 nclass 19## The minimum class member count is under the setting value.
## bic -99988.3 nclass 18## The minimum class member count is under the setting value.
## bic -100001 nclass 17plot(result.IRM, type = "Array")
plot(result.IRM, type = "FRP", nc = 3)
plot(result.IRM, type = "TRP")
Additionally, supplementary notes on the derivation of the Infinite Relational Model with Chinese restaurant process is here.
Bayesian Network Model
The Bayesian network model is a model that represents the conditional probabilities between items in a network format based on the pass rates of the items. By providing a Directed Acyclic Graph (DAG) between items externally, it calculates the conditional probabilities based on the specified graph. The igraph package is used for the analysis and representation of the network.
There are three ways to specify the graph. You can either pass a matrix-type DAG to the argument adj_matrix, pass a DAG described in a CSV file to the argument adj_file, or pass a graph-type object g used in the igraph package to the argument g.
The methods to create the matrix-type adj_matrix and the graph object g are as follows:
library(igraph)
DAG <-
matrix(
c(
"Item01", "Item02",
"Item02", "Item03",
"Item02", "Item04",
"Item03", "Item05",
"Item04", "Item05"
),
ncol = 2, byrow = T
)
## graph object
g <- igraph::graph_from_data_frame(DAG)
g## IGRAPH aea3fb0 DN-- 5 5 --
## + attr: name (v/c)
## + edges from aea3fb0 (vertex names):
## [1] Item01->Item02 Item02->Item03 Item02->Item04 Item03->Item05 Item04->Item05## Adjacency matrix
adj_mat <- as.matrix(igraph::get.adjacency(g))
print(adj_mat)## Item01 Item02 Item03 Item04 Item05
## Item01 0 1 0 0 0
## Item02 0 0 1 1 0
## Item03 0 0 0 0 1
## Item04 0 0 0 0 1
## Item05 0 0 0 0 0A CSV file with the same information as the graph above in the following format. The first line contains column names (headers) and will not be read as data.
## From,To## Item01,Item02
## Item02,Item03
## Item02,Item04
## Item03,Item05
## Item04,Item05While only one specification is sufficient, if multiple specifications are provided, they will be prioritized in the order of file, matrix, and graph object.
An example of executing BNM by providing a graph structure (DAG) is as follows:
result.BNM <- BNM(J5S10, adj_matrix = adj_mat)
result.BNM## Adjacency Matrix
## Item01 Item02 Item03 Item04 Item05
## Item01 0 1 0 0 0
## Item02 0 0 1 1 0
## Item03 0 0 0 0 1
## Item04 0 0 0 0 1
## Item05 0 0 0 0 0
## [1] "Your graph is an acyclic graph."
## [1] "Your graph is connected DAG."
##
## Parameter Learning
## PIRP 1 PIRP 2 PIRP 3 PIRP 4
## Item01 0.600
## Item02 0.250 0.5
## Item03 0.833 1.0
## Item04 0.167 0.5
## Item05 0.000 NaN 0.333 0.667
##
## Conditional Correct Response Rate
## Child Item N of Parents Parent Items PIRP Conditional CRR
## 1 Item01 0 No Parents No Pattern 0.6000000
## 2 Item02 1 Item01 0 0.2500000
## 3 Item02 1 Item01 1 0.5000000
## 4 Item03 1 Item02 0 0.8333333
## 5 Item03 1 Item02 1 1.0000000
## 6 Item04 1 Item02 0 0.1666667
## 7 Item04 1 Item02 1 0.5000000
## 8 Item05 2 Item03, Item04 00 0.0000000
## 9 Item05 2 Item03, Item04 01 NaN(0/0)
## 10 Item05 2 Item03, Item04 10 0.3333333
## 11 Item05 2 Item03, Item04 11 0.6666667
##
## Model Fit Indices
## value
## model_log_like -26.411
## bench_log_like -8.935
## null_log_like -28.882
## model_Chi_sq 34.953
## null_Chi_sq 39.894
## model_df 20.000
## null_df 25.000
## NFI 0.124
## RFI 0.000
## IFI 0.248
## TLI 0.000
## CFI 0.000
## RMSEA 0.288
## AIC -5.047
## CAIC -13.005
## BIC -11.099Structure Learning for Bayesian network with GA
The function searches for a DAG suitable for the data using a genetic algorithm. A best DAG is not necessarily identified. Instead of exploring all combinations of nodes and edges, only the space topologically sorted by the pass rate, namely the upper triangular matrix of the adjacency matrix, is explored. For interpretability, the number of parent nodes should be limited. A null model is not proposed. Utilize the content of the items and the experience of the questioner to aid in interpreting the results. For more details, please refer to Section 8.5 of the text(Shojima,2022).
Please note that the GA may take a considerable amount of time, depending on the number of items and the size of the population.
StrLearningGA_BNM(J5S10,
population = 20, Rs = 0.5, Rm = 0.002, maxParents = 2,
maxGeneration = 100, crossover = 2, elitism = 2
)## Adjacency Matrix
## Item01 Item02 Item03 Item04 Item05
## Item01 0 0 0 1 0
## Item02 0 0 0 0 0
## Item03 0 0 0 0 0
## Item04 0 0 0 0 0
## Item05 0 0 0 0 0
## [1] "Your graph is an acyclic graph."
## [1] "Your graph is connected DAG."
##
## Parameter Learning
## PIRP 1 PIRP 2
## Item01 0.6
## Item02 0.4
## Item03 0.9
## Item04 0.0 0.5
## Item05 0.4
##
## Conditional Correct Response Rate
## Child Item N of Parents Parent Items PIRP Conditional CRR
## 1 Item01 0 No Parents No Pattern 0.6000000
## 2 Item02 0 No Parents No Pattern 0.4000000
## 3 Item03 0 No Parents No Pattern 0.9000000
## 4 Item04 1 Item01 0 0.0000000
## 5 Item04 1 Item01 1 0.5000000
## 6 Item05 0 No Parents No Pattern 0.4000000
##
## Model Fit Indices
## value
## model_log_like -26.959
## bench_log_like -8.935
## null_log_like -28.882
## model_Chi_sq 36.048
## null_Chi_sq 39.894
## model_df 24.000
## null_df 25.000
## NFI 0.096
## RFI 0.059
## IFI 0.242
## TLI 0.157
## CFI 0.191
## RMSEA 0.236
## AIC -11.952
## CAIC -21.502
## BIC -19.214The method of Population-Based incremental learning proposed by Fukuda (2014) can also be used for learning. This method has several variations for estimating the optimal adjacency matrix at the end, which can be specified as options. See help or text Section 8.5.2.
StrLearningPBIL_BNM(J5S10,
population = 20, Rs = 0.5, Rm = 0.005, maxParents = 2,
alpha = 0.05, estimate = 4
)## Adjacency Matrix
## Item01 Item02 Item03 Item04 Item05
## Item01 0 0 0 1 0
## Item02 0 0 0 0 0
## Item03 0 0 0 0 0
## Item04 0 0 0 0 0
## Item05 0 0 0 0 0
## [1] "Your graph is an acyclic graph."
## [1] "Your graph is connected DAG."
##
## Parameter Learning
## PIRP 1 PIRP 2
## Item01 0.6
## Item02 0.4
## Item03 0.9
## Item04 0.0 0.5
## Item05 0.4
##
## Conditional Correct Response Rate
## Child Item N of Parents Parent Items PIRP Conditional CRR
## 1 Item01 0 No Parents No Pattern 0.6000000
## 2 Item02 0 No Parents No Pattern 0.4000000
## 3 Item03 0 No Parents No Pattern 0.9000000
## 4 Item04 1 Item01 0 0.0000000
## 5 Item04 1 Item01 1 0.5000000
## 6 Item05 0 No Parents No Pattern 0.4000000
##
## Model Fit Indices
## value
## model_log_like -26.959
## bench_log_like -8.935
## null_log_like -28.882
## model_Chi_sq 36.048
## null_Chi_sq 39.894
## model_df 24.000
## null_df 25.000
## NFI 0.096
## RFI 0.059
## IFI 0.242
## TLI 0.157
## CFI 0.191
## RMSEA 0.236
## AIC -11.952
## CAIC -21.502
## BIC -19.214Local Dependence Latent Rank Analysis
LD-LRA is an analysis that combines LRA and BNM, and it is used to analyze the network structure among items in the latent rank. In this function, structural learning is not performed, so you need to provide item graphs for each rank as separate files.
For each class, it is necessary to specify a graph, and there are three ways to do so. You can either pass a matrix-type DAG for each class or a list of graph-type objects used in the igraph package to the arguments adj_list or g_list, respectively, or you can provide a DAG described in a CSV file. The way to specify it in a CSV file is as follows.
DAG_dat <- matrix(c(
"From", "To", "Rank",
"Item01", "Item02", 1,
"Item04", "Item05", 1,
"Item01", "Item02", 2,
"Item02", "Item03", 2,
"Item04", "Item05", 2,
"Item08", "Item09", 2,
"Item08", "Item10", 2,
"Item09", "Item10", 2,
"Item08", "Item11", 2,
"Item01", "Item02", 3,
"Item02", "Item03", 3,
"Item04", "Item05", 3,
"Item08", "Item09", 3,
"Item08", "Item10", 3,
"Item09", "Item10", 3,
"Item08", "Item11", 3,
"Item02", "Item03", 4,
"Item04", "Item06", 4,
"Item04", "Item07", 4,
"Item05", "Item06", 4,
"Item05", "Item07", 4,
"Item08", "Item10", 4,
"Item08", "Item11", 4,
"Item09", "Item11", 4,
"Item02", "Item03", 5,
"Item04", "Item06", 5,
"Item04", "Item07", 5,
"Item05", "Item06", 5,
"Item05", "Item07", 5,
"Item09", "Item11", 5,
"Item10", "Item11", 5,
"Item10", "Item12", 5
), ncol = 3, byrow = TRUE)
# save csv file
edgeFile <- tempfile(fileext = ".csv")
write.csv(DAG_dat, edgeFile, row.names = FALSE, quote = TRUE)Here, it is shown an example of specifying with matrix-type and graph objects using the aforementioned CSV file. While only one specification is sufficient, if multiple specifications are provided, they will be prioritized in the order of file, matrix, and graph object.
g_csv <- read.csv(edgeFile)
colnames(g_csv) <- c("From", "To", "Rank")
adj_list <- list()
g_list <- list()
for (i in 1:5) {
adj_R <- g_csv[g_csv$Rank == i, 1:2]
g_tmp <- igraph::graph_from_data_frame(adj_R)
adj_tmp <- igraph::get.adjacency(g_tmp)
g_list[[i]] <- g_tmp
adj_list[[i]] <- adj_tmp
}
## Example of graph list
g_list## [[1]]
## IGRAPH b9ddf28 DN-- 4 2 --
## + attr: name (v/c)
## + edges from b9ddf28 (vertex names):
## [1] Item01->Item02 Item04->Item05
##
## [[2]]
## IGRAPH 67992ed DN-- 9 7 --
## + attr: name (v/c)
## + edges from 67992ed (vertex names):
## [1] Item01->Item02 Item02->Item03 Item04->Item05 Item08->Item09 Item08->Item10
## [6] Item09->Item10 Item08->Item11
##
## [[3]]
## IGRAPH 316d787 DN-- 9 7 --
## + attr: name (v/c)
## + edges from 316d787 (vertex names):
## [1] Item01->Item02 Item02->Item03 Item04->Item05 Item08->Item09 Item08->Item10
## [6] Item09->Item10 Item08->Item11
##
## [[4]]
## IGRAPH 9312809 DN-- 10 8 --
## + attr: name (v/c)
## + edges from 9312809 (vertex names):
## [1] Item02->Item03 Item04->Item06 Item04->Item07 Item05->Item06 Item05->Item07
## [6] Item08->Item10 Item08->Item11 Item09->Item11
##
## [[5]]
## IGRAPH 8f4124e DN-- 10 8 --
## + attr: name (v/c)
## + edges from 8f4124e (vertex names):
## [1] Item02->Item03 Item04->Item06 Item04->Item07 Item05->Item06 Item05->Item07
## [6] Item09->Item11 Item10->Item11 Item10->Item12### Example of adjacency list
adj_list## [[1]]
## 4 x 4 sparse Matrix of class "dgCMatrix"
## Item01 Item04 Item02 Item05
## Item01 . . 1 .
## Item04 . . . 1
## Item02 . . . .
## Item05 . . . .
##
## [[2]]
## 9 x 9 sparse Matrix of class "dgCMatrix"
## Item01 Item02 Item04 Item08 Item09 Item03 Item05 Item10 Item11
## Item01 . 1 . . . . . . .
## Item02 . . . . . 1 . . .
## Item04 . . . . . . 1 . .
## Item08 . . . . 1 . . 1 1
## Item09 . . . . . . . 1 .
## Item03 . . . . . . . . .
## Item05 . . . . . . . . .
## Item10 . . . . . . . . .
## Item11 . . . . . . . . .
##
## [[3]]
## 9 x 9 sparse Matrix of class "dgCMatrix"
## Item01 Item02 Item04 Item08 Item09 Item03 Item05 Item10 Item11
## Item01 . 1 . . . . . . .
## Item02 . . . . . 1 . . .
## Item04 . . . . . . 1 . .
## Item08 . . . . 1 . . 1 1
## Item09 . . . . . . . 1 .
## Item03 . . . . . . . . .
## Item05 . . . . . . . . .
## Item10 . . . . . . . . .
## Item11 . . . . . . . . .
##
## [[4]]
## 10 x 10 sparse Matrix of class "dgCMatrix"##
## Item02 . . . . . 1 . . . .
## Item04 . . . . . . 1 1 . .
## Item05 . . . . . . 1 1 . .
## Item08 . . . . . . . . 1 1
## Item09 . . . . . . . . . 1
## Item03 . . . . . . . . . .
## Item06 . . . . . . . . . .
## Item07 . . . . . . . . . .
## Item10 . . . . . . . . . .
## Item11 . . . . . . . . . .
##
## [[5]]
## 10 x 10 sparse Matrix of class "dgCMatrix"##
## Item02 . . . . . 1 . . . .
## Item04 . . . . . . 1 1 . .
## Item05 . . . . . . 1 1 . .
## Item09 . . . . . . . . 1 .
## Item10 . . . . . . . . 1 1
## Item03 . . . . . . . . . .
## Item06 . . . . . . . . . .
## Item07 . . . . . . . . . .
## Item11 . . . . . . . . . .
## Item12 . . . . . . . . . .The example of running the LDLRA function using this CSV file would look like this.
result.LDLRA <- LDLRA(J12S5000,
ncls = 5,
adj_file = edgeFile
)
result.LDLRA## Adjacency Matrix
## [[1]]
## Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10 Item11
## Item01 0 1 0 0 0 0 0 0 0 0 0
## Item02 0 0 0 0 0 0 0 0 0 0 0
## Item03 0 0 0 0 0 0 0 0 0 0 0
## Item04 0 0 0 0 1 0 0 0 0 0 0
## Item05 0 0 0 0 0 0 0 0 0 0 0
## Item06 0 0 0 0 0 0 0 0 0 0 0
## Item07 0 0 0 0 0 0 0 0 0 0 0
## Item08 0 0 0 0 0 0 0 0 0 0 0
## Item09 0 0 0 0 0 0 0 0 0 0 0
## Item10 0 0 0 0 0 0 0 0 0 0 0
## Item11 0 0 0 0 0 0 0 0 0 0 0
## Item12 0 0 0 0 0 0 0 0 0 0 0
## Item12
## Item01 0
## Item02 0
## Item03 0
## Item04 0
## Item05 0
## Item06 0
## Item07 0
## Item08 0
## Item09 0
## Item10 0
## Item11 0
## Item12 0
##
## [[2]]
## Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10 Item11
## Item01 0 1 0 0 0 0 0 0 0 0 0
## Item02 0 0 1 0 0 0 0 0 0 0 0
## Item03 0 0 0 0 0 0 0 0 0 0 0
## Item04 0 0 0 0 1 0 0 0 0 0 0
## Item05 0 0 0 0 0 0 0 0 0 0 0
## Item06 0 0 0 0 0 0 0 0 0 0 0
## Item07 0 0 0 0 0 0 0 0 0 0 0
## Item08 0 0 0 0 0 0 0 0 1 1 1
## Item09 0 0 0 0 0 0 0 0 0 1 0
## Item10 0 0 0 0 0 0 0 0 0 0 0
## Item11 0 0 0 0 0 0 0 0 0 0 0
## Item12 0 0 0 0 0 0 0 0 0 0 0
## Item12
## Item01 0
## Item02 0
## Item03 0
## Item04 0
## Item05 0
## Item06 0
## Item07 0
## Item08 0
## Item09 0
## Item10 0
## Item11 0
## Item12 0
##
## [[3]]
## Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10 Item11
## Item01 0 1 0 0 0 0 0 0 0 0 0
## Item02 0 0 1 0 0 0 0 0 0 0 0
## Item03 0 0 0 0 0 0 0 0 0 0 0
## Item04 0 0 0 0 1 0 0 0 0 0 0
## Item05 0 0 0 0 0 0 0 0 0 0 0
## Item06 0 0 0 0 0 0 0 0 0 0 0
## Item07 0 0 0 0 0 0 0 0 0 0 0
## Item08 0 0 0 0 0 0 0 0 1 1 1
## Item09 0 0 0 0 0 0 0 0 0 1 0
## Item10 0 0 0 0 0 0 0 0 0 0 0
## Item11 0 0 0 0 0 0 0 0 0 0 0
## Item12 0 0 0 0 0 0 0 0 0 0 0
## Item12
## Item01 0
## Item02 0
## Item03 0
## Item04 0
## Item05 0
## Item06 0
## Item07 0
## Item08 0
## Item09 0
## Item10 0
## Item11 0
## Item12 0
##
## [[4]]
## Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10 Item11
## Item01 0 0 0 0 0 0 0 0 0 0 0
## Item02 0 0 1 0 0 0 0 0 0 0 0
## Item03 0 0 0 0 0 0 0 0 0 0 0
## Item04 0 0 0 0 0 1 1 0 0 0 0
## Item05 0 0 0 0 0 1 1 0 0 0 0
## Item06 0 0 0 0 0 0 0 0 0 0 0
## Item07 0 0 0 0 0 0 0 0 0 0 0
## Item08 0 0 0 0 0 0 0 0 0 1 1
## Item09 0 0 0 0 0 0 0 0 0 0 1
## Item10 0 0 0 0 0 0 0 0 0 0 0
## Item11 0 0 0 0 0 0 0 0 0 0 0
## Item12 0 0 0 0 0 0 0 0 0 0 0
## Item12
## Item01 0
## Item02 0
## Item03 0
## Item04 0
## Item05 0
## Item06 0
## Item07 0
## Item08 0
## Item09 0
## Item10 0
## Item11 0
## Item12 0
##
## [[5]]
## Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10 Item11
## Item01 0 0 0 0 0 0 0 0 0 0 0
## Item02 0 0 1 0 0 0 0 0 0 0 0
## Item03 0 0 0 0 0 0 0 0 0 0 0
## Item04 0 0 0 0 0 1 1 0 0 0 0
## Item05 0 0 0 0 0 1 1 0 0 0 0
## Item06 0 0 0 0 0 0 0 0 0 0 0
## Item07 0 0 0 0 0 0 0 0 0 0 0
## Item08 0 0 0 0 0 0 0 0 0 0 0
## Item09 0 0 0 0 0 0 0 0 0 0 1
## Item10 0 0 0 0 0 0 0 0 0 0 1
## Item11 0 0 0 0 0 0 0 0 0 0 0
## Item12 0 0 0 0 0 0 0 0 0 0 0
## Item12
## Item01 0
## Item02 0
## Item03 0
## Item04 0
## Item05 0
## Item06 0
## Item07 0
## Item08 0
## Item09 0
## Item10 1
## Item11 0
## Item12 0
##
## Parameter Learning
## Item Rank PIRP 1 PIRP 2 PIRP 3 PIRP 4
## 1 Item01 1 0.456
## 2 Item02 1 0.030 0.444
## 3 Item03 1 0.083
## 4 Item04 1 0.421
## 5 Item05 1 0.101 0.240
## 6 Item06 1 0.025
## 7 Item07 1 0.016
## 8 Item08 1 0.286
## 9 Item09 1 0.326
## 10 Item10 1 0.181
## 11 Item11 1 0.106
## 12 Item12 1 0.055
## 13 Item01 2 0.549
## 14 Item02 2 0.035 0.568
## 15 Item03 2 0.020 0.459
## 16 Item04 2 0.495
## 17 Item05 2 0.148 0.351
## 18 Item06 2 0.066
## 19 Item07 2 0.045
## 20 Item08 2 0.407
## 21 Item09 2 0.264 0.734
## 22 Item10 2 0.081 0.133 0.159 0.745
## 23 Item11 2 0.041 0.445
## 24 Item12 2 0.086
## 25 Item01 3 0.683
## 26 Item02 3 0.040 0.728
## 27 Item03 3 0.032 0.617
## 28 Item04 3 0.612
## 29 Item05 3 0.227 0.556
## 30 Item06 3 0.205
## 31 Item07 3 0.156
## 32 Item08 3 0.581
## 33 Item09 3 0.330 0.845
## 34 Item10 3 0.092 0.160 0.211 0.843
## 35 Item11 3 0.056 0.636
## 36 Item12 3 0.152
## 37 Item01 4 0.836
## 38 Item02 4 0.720
## 39 Item03 4 0.058 0.713
## 40 Item04 4 0.740
## 41 Item05 4 0.635
## 42 Item06 4 0.008 0.105 0.023 0.684
## 43 Item07 4 0.010 0.031 0.039 0.542
## 44 Item08 4 0.760
## 45 Item09 4 0.805
## 46 Item10 4 0.150 0.844
## 47 Item11 4 0.064 0.124 0.105 0.825
## 48 Item12 4 0.227
## 49 Item01 5 0.931
## 50 Item02 5 0.869
## 51 Item03 5 0.099 0.789
## 52 Item04 5 0.846
## 53 Item05 5 0.811
## 54 Item06 5 0.015 0.125 0.040 0.788
## 55 Item07 5 0.016 0.034 0.064 0.650
## 56 Item08 5 0.880
## 57 Item09 5 0.912
## 58 Item10 5 0.825
## 59 Item11 5 0.082 0.190 0.216 0.915
## 60 Item12 5 0.153 0.341
##
## Conditional Correct Response Rate
## Child Item Rank N of Parents Parent Items PIRP Conditional CRR
## 1 Item01 1 0 No Parents No Pattern 0.45558
## 2 Item02 1 1 Item01 0 0.03025
## 3 Item02 1 1 Item01 1 0.44394
## 4 Item03 1 0 No Parents No Pattern 0.08278
## 5 Item04 1 0 No Parents No Pattern 0.42148
## 6 Item05 1 1 Item04 0 0.10127
## 7 Item05 1 1 Item04 1 0.24025
## 8 Item06 1 0 No Parents No Pattern 0.02499
## 9 Item07 1 0 No Parents No Pattern 0.01574
## 10 Item08 1 0 No Parents No Pattern 0.28642
## 11 Item09 1 0 No Parents No Pattern 0.32630
## 12 Item10 1 0 No Parents No Pattern 0.18092
## 13 Item11 1 0 No Parents No Pattern 0.10575
## 14 Item12 1 0 No Parents No Pattern 0.05523
## 15 Item01 2 0 No Parents No Pattern 0.54940
## 16 Item02 2 1 Item01 0 0.03471
## 17 Item02 2 1 Item01 1 0.56821
## 18 Item03 2 1 Item02 0 0.02016
## 19 Item03 2 1 Item02 1 0.45853
## 20 Item04 2 0 No Parents No Pattern 0.49508
## 21 Item05 2 1 Item04 0 0.14771
## 22 Item05 2 1 Item04 1 0.35073
## 23 Item06 2 0 No Parents No Pattern 0.06647
## 24 Item07 2 0 No Parents No Pattern 0.04491
## 25 Item08 2 0 No Parents No Pattern 0.40721
## 26 Item09 2 1 Item08 0 0.26431
## 27 Item09 2 1 Item08 1 0.73427
## 28 Item10 2 2 Item08, Item09 00 0.08098
## 29 Item10 2 2 Item08, Item09 01 0.13279
## 30 Item10 2 2 Item08, Item09 10 0.15937
## 31 Item10 2 2 Item08, Item09 11 0.74499
## 32 Item11 2 1 Item08 0 0.04094
## 33 Item11 2 1 Item08 1 0.44457
## 34 Item12 2 0 No Parents No Pattern 0.08574
## 35 Item01 3 0 No Parents No Pattern 0.68342
## 36 Item02 3 1 Item01 0 0.04020
## 37 Item02 3 1 Item01 1 0.72757
## 38 Item03 3 1 Item02 0 0.03175
## 39 Item03 3 1 Item02 1 0.61691
## 40 Item04 3 0 No Parents No Pattern 0.61195
## 41 Item05 3 1 Item04 0 0.22705
## 42 Item05 3 1 Item04 1 0.55588
## 43 Item06 3 0 No Parents No Pattern 0.20488
## 44 Item07 3 0 No Parents No Pattern 0.15633
## 45 Item08 3 0 No Parents No Pattern 0.58065
## 46 Item09 3 1 Item08 0 0.32967
## 47 Item09 3 1 Item08 1 0.84549
## 48 Item10 3 2 Item08, Item09 00 0.09192
## 49 Item10 3 2 Item08, Item09 01 0.15977
## 50 Item10 3 2 Item08, Item09 10 0.21087
## 51 Item10 3 2 Item08, Item09 11 0.84330
## 52 Item11 3 1 Item08 0 0.05581
## 53 Item11 3 1 Item08 1 0.63598
## 54 Item12 3 0 No Parents No Pattern 0.15169
## 55 Item01 4 0 No Parents No Pattern 0.83557
## 56 Item02 4 0 No Parents No Pattern 0.71950
## 57 Item03 4 1 Item02 0 0.05808
## 58 Item03 4 1 Item02 1 0.71297
## 59 Item04 4 0 No Parents No Pattern 0.73957
## 60 Item05 4 0 No Parents No Pattern 0.63526
## 61 Item06 4 2 Item04, Item05 00 0.00816
## 62 Item06 4 2 Item04, Item05 01 0.10474
## 63 Item06 4 2 Item04, Item05 10 0.02265
## 64 Item06 4 2 Item04, Item05 11 0.68419
## 65 Item07 4 2 Item04, Item05 00 0.00984
## 66 Item07 4 2 Item04, Item05 01 0.03091
## 67 Item07 4 2 Item04, Item05 10 0.03850
## 68 Item07 4 2 Item04, Item05 11 0.54195
## 69 Item08 4 0 No Parents No Pattern 0.75976
## 70 Item09 4 0 No Parents No Pattern 0.80490
## 71 Item10 4 1 Item08 0 0.14956
## 72 Item10 4 1 Item08 1 0.84430
## 73 Item11 4 2 Item08, Item09 00 0.06376
## 74 Item11 4 2 Item08, Item09 01 0.12384
## 75 Item11 4 2 Item08, Item09 10 0.10494
## 76 Item11 4 2 Item08, Item09 11 0.82451
## 77 Item12 4 0 No Parents No Pattern 0.22688
## 78 Item01 5 0 No Parents No Pattern 0.93131
## 79 Item02 5 0 No Parents No Pattern 0.86923
## 80 Item03 5 1 Item02 0 0.09865
## 81 Item03 5 1 Item02 1 0.78854
## 82 Item04 5 0 No Parents No Pattern 0.84621
## 83 Item05 5 0 No Parents No Pattern 0.81118
## 84 Item06 5 2 Item04, Item05 00 0.01452
## 85 Item06 5 2 Item04, Item05 01 0.12528
## 86 Item06 5 2 Item04, Item05 10 0.04000
## 87 Item06 5 2 Item04, Item05 11 0.78805
## 88 Item07 5 2 Item04, Item05 00 0.01570
## 89 Item07 5 2 Item04, Item05 01 0.03361
## 90 Item07 5 2 Item04, Item05 10 0.06363
## 91 Item07 5 2 Item04, Item05 11 0.65039
## 92 Item08 5 0 No Parents No Pattern 0.88028
## 93 Item09 5 0 No Parents No Pattern 0.91209
## 94 Item10 5 0 No Parents No Pattern 0.82476
## 95 Item11 5 2 Item09, Item10 00 0.08248
## 96 Item11 5 2 Item09, Item10 01 0.18951
## 97 Item11 5 2 Item09, Item10 10 0.21590
## 98 Item11 5 2 Item09, Item10 11 0.91466
## 99 Item12 5 1 Item10 0 0.15301
## 100 Item12 5 1 Item10 1 0.34114
##
## Marginal Item Reference Profile
## Rank 1 Rank 2 Rank 3 Rank 4 Rank 5
## Item01 0.4556 0.5494 0.683 0.836 0.931
## Item02 0.2099 0.2964 0.474 0.720 0.869
## Item03 0.0828 0.1397 0.316 0.554 0.741
## Item04 0.4215 0.4951 0.612 0.740 0.846
## Item05 0.1555 0.2393 0.432 0.635 0.811
## Item06 0.0250 0.0665 0.205 0.385 0.631
## Item07 0.0157 0.0449 0.156 0.304 0.517
## Item08 0.2864 0.4072 0.581 0.760 0.880
## Item09 0.3263 0.4409 0.624 0.805 0.912
## Item10 0.1809 0.2977 0.498 0.650 0.825
## Item11 0.1057 0.1926 0.387 0.565 0.808
## Item12 0.0552 0.0857 0.152 0.227 0.317
##
## IRP Indices
## Alpha A Beta B Gamma C
## Item01 3 0.15215133 1 0.4555806 0 0
## Item02 3 0.24578705 3 0.4737140 0 0
## Item03 3 0.23808314 4 0.5544465 0 0
## Item04 3 0.12762155 2 0.4950757 0 0
## Item05 3 0.20322441 3 0.4320364 0 0
## Item06 4 0.24595102 4 0.3851075 0 0
## Item07 4 0.21361675 5 0.5173874 0 0
## Item08 3 0.17910918 3 0.5806476 0 0
## Item09 2 0.18320368 2 0.4408936 0 0
## Item10 2 0.20070396 3 0.4984108 0 0
## Item11 4 0.24332189 4 0.5650492 0 0
## Item12 4 0.09047482 5 0.3173548 0 0
## [1] "Strongly ordinal alignment condition was satisfied."
##
## Test reference Profile and Latent Rank Distribution
## Rank 1 Rank 2 Rank 3 Rank 4 Rank 5
## Test Reference Profile 2.321 3.255 5.121 7.179 9.090
## Latent Rank Ditribution 1829.000 593.000 759.000 569.000 1250.000
## Rank Membership Distribution 1121.838 1087.855 873.796 835.528 1080.983
## [1] "Weakly ordinal alignment condition was satisfied."
##
## Model Fit Indices
## value
## model_log_like -26657.783
## bench_log_like -21318.465
## null_log_like -37736.228
## model_Chi_sq 10678.636
## null_Chi_sq 32835.527
## model_df 56.000
## null_df 144.000
## NFI 0.675
## RFI 0.164
## IFI 0.676
## TLI 0.164
## CFI 0.675
## RMSEA 0.195
## AIC 10566.636
## CAIC 10201.662
## BIC 10201.673Of course, it also supports various types of plots.
plot(result.LDLRA, type = "IRP", nc = 4, nr = 3)
plot(result.LDLRA, type = "TRP")
plot(result.LDLRA, type = "LRD")
Structure Learning for LDLRA with GA(PBIL)
You can learn item-interaction graphs for each rank using the PBIL algorithm. In addition to various options, the learning process requires a very long computation time. It’s also important to note that the result is merely one of the feasible solutions, and it’s not necessarily the optimal solution.
result.LDLRA.PBIL <- StrLearningPBIL_LDLRA(J35S515,
seed = 123,
ncls = 5,
method = "R",
elitism = 1,
successiveLimit = 15
)
result.LDLRA.PBILLocal Dependence Biclustering
Local Dependence Biclustering combines biclustering and Bayesian network models. The model requires three main components:
- Number of latent classes/ranks
- Field assignments for items
- Network structure between fields at each rank
Here’s an example implementation:
# Create field configuration vector (assign items to fields)
conf <- c(1, 6, 6, 8, 9, 9, 4, 7, 7, 7, 5, 8, 9, 10, 10, 9, 9, 10, 10, 10, 2, 2, 3, 3, 5, 5, 6, 9, 9, 10, 1, 1, 7, 9, 10)
# Create edge data for network structure between fields
edges_data <- data.frame(
"From Field (Parent) >>>" = c(
6, 4, 5, 1, 1, 4, # Class/Rank 2
3, 4, 6, 2, 4, 4, # Class/Rank 3
3, 6, 4, 1, # Class/Rank 4
7, 9, 6, 7 # Class/Rank 5
),
">>> To Field (Child)" = c(
8, 7, 8, 7, 2, 5, # Class/Rank 2
5, 8, 8, 4, 6, 7, # Class/Rank 3
5, 8, 5, 8, # Class/Rank 4
10, 10, 8, 9 # Class/Rank 5
),
"At Class/Rank (Locus)" = c(
2, 2, 2, 2, 2, 2, # Class/Rank 2
3, 3, 3, 3, 3, 3, # Class/Rank 3
4, 4, 4, 4, # Class/Rank 4
5, 5, 5, 5 # Class/Rank 5
)
)
# Save edge data to temporary file
edgeFile <- tempfile(fileext = ".csv")
write.csv(edges_data, file = edgeFile, row.names = FALSE)Additionally, as mentioned in the text (Shojima, 2022), it is often the case that seeking the network structure exploratively does not yield appropriate results, so it has not been implemented.
result.LDB <- LDB(U = J35S515, ncls = 5, conf = conf, adj_file = edgeFile)
result.LDB## Adjacency Matrix
## [[1]]
## Field01 Field02 Field03 Field04 Field05 Field06 Field07 Field08 Field09
## Field01 0 0 0 0 0 0 0 0 0
## Field02 0 0 0 0 0 0 0 0 0
## Field03 0 0 0 0 0 0 0 0 0
## Field04 0 0 0 0 0 0 0 0 0
## Field05 0 0 0 0 0 0 0 0 0
## Field06 0 0 0 0 0 0 0 0 0
## Field07 0 0 0 0 0 0 0 0 0
## Field08 0 0 0 0 0 0 0 0 0
## Field09 0 0 0 0 0 0 0 0 0
## Field10 0 0 0 0 0 0 0 0 0
## Field10
## Field01 0
## Field02 0
## Field03 0
## Field04 0
## Field05 0
## Field06 0
## Field07 0
## Field08 0
## Field09 0
## Field10 0
##
## [[2]]
## Field01 Field02 Field03 Field04 Field05 Field06 Field07 Field08 Field09
## Field01 0 1 0 0 0 0 1 0 0
## Field02 0 0 0 0 0 0 0 0 0
## Field03 0 0 0 0 0 0 0 0 0
## Field04 0 0 0 0 1 0 1 0 0
## Field05 0 0 0 0 0 0 0 1 0
## Field06 0 0 0 0 0 0 0 1 0
## Field07 0 0 0 0 0 0 0 0 0
## Field08 0 0 0 0 0 0 0 0 0
## Field09 0 0 0 0 0 0 0 0 0
## Field10 0 0 0 0 0 0 0 0 0
## Field10
## Field01 0
## Field02 0
## Field03 0
## Field04 0
## Field05 0
## Field06 0
## Field07 0
## Field08 0
## Field09 0
## Field10 0
##
## [[3]]
## Field01 Field02 Field03 Field04 Field05 Field06 Field07 Field08 Field09
## Field01 0 0 0 0 0 0 0 0 0
## Field02 0 0 0 1 0 0 0 0 0
## Field03 0 0 0 0 1 0 0 0 0
## Field04 0 0 0 0 0 1 1 1 0
## Field05 0 0 0 0 0 0 0 0 0
## Field06 0 0 0 0 0 0 0 1 0
## Field07 0 0 0 0 0 0 0 0 0
## Field08 0 0 0 0 0 0 0 0 0
## Field09 0 0 0 0 0 0 0 0 0
## Field10 0 0 0 0 0 0 0 0 0
## Field10
## Field01 0
## Field02 0
## Field03 0
## Field04 0
## Field05 0
## Field06 0
## Field07 0
## Field08 0
## Field09 0
## Field10 0
##
## [[4]]
## Field01 Field02 Field03 Field04 Field05 Field06 Field07 Field08 Field09
## Field01 0 0 0 0 0 0 0 1 0
## Field02 0 0 0 0 0 0 0 0 0
## Field03 0 0 0 0 1 0 0 0 0
## Field04 0 0 0 0 1 0 0 0 0
## Field05 0 0 0 0 0 0 0 0 0
## Field06 0 0 0 0 0 0 0 1 0
## Field07 0 0 0 0 0 0 0 0 0
## Field08 0 0 0 0 0 0 0 0 0
## Field09 0 0 0 0 0 0 0 0 0
## Field10 0 0 0 0 0 0 0 0 0
## Field10
## Field01 0
## Field02 0
## Field03 0
## Field04 0
## Field05 0
## Field06 0
## Field07 0
## Field08 0
## Field09 0
## Field10 0
##
## [[5]]
## Field01 Field02 Field03 Field04 Field05 Field06 Field07 Field08 Field09
## Field01 0 0 0 0 0 0 0 0 0
## Field02 0 0 0 0 0 0 0 0 0
## Field03 0 0 0 0 0 0 0 0 0
## Field04 0 0 0 0 0 0 0 0 0
## Field05 0 0 0 0 0 0 0 0 0
## Field06 0 0 0 0 0 0 0 1 0
## Field07 0 0 0 0 0 0 0 0 1
## Field08 0 0 0 0 0 0 0 0 0
## Field09 0 0 0 0 0 0 0 0 0
## Field10 0 0 0 0 0 0 0 0 0
## Field10
## Field01 0
## Field02 0
## Field03 0
## Field04 0
## Field05 0
## Field06 0
## Field07 1
## Field08 0
## Field09 1
## Field10 0
##
## Parameter Learning
## Rank 1
## PIRP 0 PIRP 1 PIRP 2 PIRP 3 PIRP 4 PIRP 5 PIRP 6 PIRP 7 PIRP 8 PIRP 9
## Field01 0.6538
## Field02 0.0756
## Field03 0.1835
## Field04 0.3819
## Field05 0.0500
## Field06 0.0985
## Field07 0.2176
## Field08 0.0608
## Field09 0.0563
## Field10 0.0237
## PIRP 10 PIRP 11 PIRP 12
## Field01
## Field02
## Field03
## Field04
## Field05
## Field06
## Field07
## Field08
## Field09
## Field10
## Rank 2
## PIRP 0 PIRP 1 PIRP 2 PIRP 3 PIRP 4 PIRP 5 PIRP 6 PIRP 7 PIRP 8 PIRP 9
## Field01 0.8216
## Field02 0.1463 0.3181 0.383 0.597
## Field03 0.3320
## Field04 0.4931
## Field05 0.1596 0.2552
## Field06 0.2541
## Field07 0.1232 0.2926 0.217 0.306 0.376
## Field08 0.0648 0.0887 0.236 0.443 0.196 0.285 0.624
## Field09 0.1101
## Field10 0.0359
## PIRP 10 PIRP 11 PIRP 12
## Field01
## Field02
## Field03
## Field04
## Field05
## Field06
## Field07
## Field08
## Field09
## Field10
## Rank 3
## PIRP 0 PIRP 1 PIRP 2 PIRP 3 PIRP 4 PIRP 5 PIRP 6 PIRP 7 PIRP 8 PIRP 9
## Field01 0.8923
## Field02 0.8736
## Field03 0.8030
## Field04 0.4730 0.492 0.650
## Field05 0.2732 0.319 0.714
## Field06 0.4025 0.486
## Field07 0.3162 0.408
## Field08 0.1028 0.166 0.177 0.439 0.59
## Field09 0.1799
## Field10 0.0431
## PIRP 10 PIRP 11 PIRP 12
## Field01
## Field02
## Field03
## Field04
## Field05
## Field06
## Field07
## Field08
## Field09
## Field10
## Rank 4
## PIRP 0 PIRP 1 PIRP 2 PIRP 3 PIRP 4 PIRP 5 PIRP 6 PIRP 7 PIRP 8 PIRP 9
## Field01 0.91975
## Field02 0.97126
## Field03 0.96955
## Field04 0.70098
## Field05 0.28691 0.476702 0.911 0.952
## Field06 0.72620
## Field07 0.48152
## Field08 0.00353 0.000122 0.370 0.370 0.401 0.532 0.779
## Field09 0.36220
## Field10 0.08630
## PIRP 10 PIRP 11 PIRP 12
## Field01
## Field02
## Field03
## Field04
## Field05
## Field06
## Field07
## Field08
## Field09
## Field10
## Rank 5
## PIRP 0 PIRP 1 PIRP 2 PIRP 3 PIRP 4 PIRP 5 PIRP 6 PIRP 7 PIRP 8 PIRP 9
## Field01 0.9627
## Field02 0.9959
## Field03 0.9947
## Field04 0.8654
## Field05 0.9939
## Field06 0.9178
## Field07 0.7334
## Field08 0.5109 0.4442 0.5939 0.9174
## Field09 0.4062 0.5193 0.6496 0.6786 0.851
## Field10 0.0874 0.0278 0.0652 0.0429 0.110 0.117 0.118 0.163 0.217 0.275
## PIRP 10 PIRP 11 PIRP 12
## Field01
## Field02
## Field03
## Field04
## Field05
## Field06
## Field07
## Field08
## Field09
## Field10 0.262 0.257 0.95
##
## Marginal Rankluster Reference Matrix
## Rank 1 Rank 2 Rank 3 Rank 4 Rank 5
## Field01 0.6538 0.8216 0.8923 0.9198 0.963
## Field02 0.0756 0.5069 0.8736 0.9713 0.996
## Field03 0.1835 0.3320 0.8030 0.9696 0.995
## Field04 0.3819 0.4931 0.6271 0.7010 0.865
## Field05 0.0500 0.2072 0.6182 0.9263 0.994
## Field06 0.0985 0.2541 0.4550 0.7262 0.918
## Field07 0.2176 0.3119 0.3738 0.4815 0.733
## Field08 0.0608 0.1723 0.2718 0.5700 0.863
## Field09 0.0563 0.1101 0.1799 0.3622 0.715
## Field10 0.0237 0.0359 0.0431 0.0863 0.377
##
## IRP Indices
## Alpha A Beta B Gamma C
## Field01 1 0.1677977 1 0.6538429 0 0
## Field02 1 0.4312713 2 0.5068824 0 0
## Field03 2 0.4710088 2 0.3320336 0 0
## Field04 4 0.1643891 2 0.4930958 0 0
## Field05 2 0.4110466 3 0.6182062 0 0
## Field06 3 0.2712108 3 0.4549879 0 0
## Field07 4 0.2518684 4 0.4815211 0 0
## Field08 3 0.2982121 4 0.5699954 0 0
## Field09 4 0.3528379 4 0.3621986 0 0
## Field10 4 0.2906998 5 0.3769977 0 0
## Rank 1 Rank 2 Rank 3 Rank 4 Rank 5
## Test Reference Profile 4.915 8.744 13.657 18.867 26.488
## Latent Rank Ditribution 163.000 91.000 102.000 91.000 68.000
## Rank Membership Dsitribution 148.275 103.002 105.606 86.100 72.017
##
## Latent Field Distribution
## Field 1 Field 2 Field 3 Field 4 Field 5 Field 6 Field 7 Field 8 Field 9
## N of Items 3 2 2 1 3 3 4 2 8
## Field 10
## N of Items 7
##
## Model Fit Indices
## value
## model_log_like -6804.899
## bench_log_like -5891.314
## null_log_like -9862.114
## model_Chi_sq 1827.169
## null_Chi_sq 7941.601
## model_df 1088.000
## null_df 1155.000
## NFI 0.770
## RFI 0.756
## IFI 0.892
## TLI 0.884
## CFI 0.891
## RMSEA 0.036
## AIC -348.831
## CAIC -4968.595
## BIC -4966.485## Strongly ordinal alignment condition was satisfied.Of course, it also supports various types of plots.
# Show bicluster structure
plot(result.LDB, type = "Array")
# Test Response Profile
plot(result.LDB, type = "TRP")
# Latent Rank Distribution
plot(result.LDB, type = "LRD")
# Rank Membership Profiles for first 9 students
plot(result.LDB, type = "RMP", students = 1:9, nc = 3, nr = 3)
# Field Reference Profiles
plot(result.LDB, type = "FRP", nc = 3, nr = 2)
In this model, you can draw a Field PIRP Profile that visualizes the correct answer count for each rank and each field.
plot(result.LDB, type = "FieldPIRP")
Bicluster Network Model
Bicluster Network Model: BINET is a model that combines the Bayesian network model and Biclustering. BINET is very similar to LDB and LDR.
The most significant difference is that in LDB, the nodes represent the fields, whereas in BINET, they represent the class. BINET explores the local dependency structure among latent classes at each latent field, where each field is a locus.
To execute this analysis, in addition to the dataset, the same field correspondence file used during exploratory Biclustering is required, as well as an adjacency matrix between classes.
# Create field configuration vector for item assignment
conf <- c(1, 5, 5, 5, 9, 9, 6, 6, 6, 6, 2, 7, 7, 11, 11, 7, 7, 12, 12, 12, 2, 2, 3, 3, 4, 4, 4, 8, 8, 12, 1, 1, 6, 10, 10)
# Create edge data for network structure between classes
edges_data <- data.frame(
"From Class (Parent) >>>" = c(
1, 2, 3, 4, 5, 7, # Dependencies in various fields
2, 4, 6, 8, 10,
6, 6, 11, 8, 9, 12
),
">>> To Class (Child)" = c(
2, 4, 5, 5, 6, 11, # Target classes
3, 7, 9, 12, 12,
10, 8, 12, 12, 11, 13
),
"At Field (Locus)" = c(
1, 2, 2, 3, 4, 4, # Field locations
5, 5, 5, 5, 5,
7, 8, 8, 9, 9, 12
)
)
# Save edge data to temporary file
edgeFile <- tempfile(fileext = ".csv")
write.csv(edges_data, file = edgeFile, row.names = FALSE)The model requires three components:
- Field assignments for items (conf vector)
- Network structure between classes for each field
- Number of classes and fields
# Fit Bicluster Network Model
result.BINET <- BINET(
U = J35S515,
ncls = 13, # Maximum class number from edges (13)
nfld = 12, # Maximum field number from conf (12)
conf = conf, # Field configuration vector
adj_file = edgeFile # Network structure file
)
# Display model results
print(result.BINET)## Total Graph
## Class01 Class02 Class03 Class04 Class05 Class06 Class07 Class08 Class09
## Class01 0 1 0 0 0 0 0 0 0
## Class02 0 0 1 1 0 0 0 0 0
## Class03 0 0 0 0 1 0 0 0 0
## Class04 0 0 0 0 1 0 1 0 0
## Class05 0 0 0 0 0 1 0 0 0
## Class06 0 0 0 0 0 0 0 1 1
## Class07 0 0 0 0 0 0 0 0 0
## Class08 0 0 0 0 0 0 0 0 0
## Class09 0 0 0 0 0 0 0 0 0
## Class10 0 0 0 0 0 0 0 0 0
## Class11 0 0 0 0 0 0 0 0 0
## Class12 0 0 0 0 0 0 0 0 0
## Class13 0 0 0 0 0 0 0 0 0
## Class10 Class11 Class12 Class13
## Class01 0 0 0 0
## Class02 0 0 0 0
## Class03 0 0 0 0
## Class04 0 0 0 0
## Class05 0 0 0 0
## Class06 1 0 0 0
## Class07 0 1 0 0
## Class08 0 0 1 0
## Class09 0 1 0 0
## Class10 0 0 1 0
## Class11 0 0 1 0
## Class12 0 0 0 1
## Class13 0 0 0 0
## Estimation of Parameter set
## Field 1
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.000
## Class 2 0.554 0.558 0.649
## Class 3 0.740
## Class 4 0.859
## Class 5 0.875
## Class 6 0.910
## Class 7 0.868
## Class 8 0.889
## Class 9 0.961
## Class 10 0.932
## Class 11 0.898
## Class 12 0.975
## Class 13 1.000
## Field 2
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.0000
## Class 2 0.0090
## Class 3 0.0396
## Class 4 0.6813 0.785 0.637
## Class 5 0.4040 0.728 0.696
## Class 6 0.6877
## Class 7 0.8316
## Class 8 0.8218
## Class 9 1.0000
## Class 10 0.9836
## Class 11 1.0000
## Class 12 1.0000
## Class 13 1.0000
## Field 3
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.000
## Class 2 0.177
## Class 3 0.219
## Class 4 0.206
## Class 5 0.189 0.253
## Class 6 1.000
## Class 7 1.000
## Class 8 1.000
## Class 9 0.986
## Class 10 1.000
## Class 11 0.973
## Class 12 1.000
## Class 13 1.000
## Field 4
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.0000
## Class 2 0.0127
## Class 3 0.1228
## Class 4 0.0468
## Class 5 0.1131
## Class 6 0.6131 0.436 0.179
## Class 7 0.9775
## Class 8 0.9539
## Class 9 0.9751
## Class 10 0.9660
## Class 11 0.9411 0.925 0.757
## Class 12 1.0000
## Class 13 1.0000
## Field 5
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.0000
## Class 2 0.0157
## Class 3 0.0731 0.330 0.06789
## Class 4 0.9626
## Class 5 0.1028
## Class 6 0.2199
## Class 7 0.1446 0.265 0.00602
## Class 8 0.9403
## Class 9 0.2936 0.298 0.12080
## Class 10 0.8255
## Class 11 0.9123
## Class 12 1.0000 1.000 1.00000
## Class 13 1.0000
## Field 6
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.000
## Class 2 0.236
## Class 3 0.275
## Class 4 0.449
## Class 5 0.414
## Class 6 0.302
## Class 7 0.415
## Class 8 0.469
## Class 9 0.560
## Class 10 0.564
## Class 11 0.614
## Class 12 0.764
## Class 13 1.000
## Field 7
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.0000
## Class 2 0.0731
## Class 3 0.0810
## Class 4 0.1924
## Class 5 0.1596
## Class 6 0.1316
## Class 7 0.1263
## Class 8 0.1792
## Class 9 0.7542
## Class 10 0.9818 0.883 0.933 0.975
## Class 11 0.3047
## Class 12 0.7862
## Class 13 1.0000
## Field 8
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.00e+00
## Class 2 9.83e-05
## Class 3 3.70e-02
## Class 4 3.91e-02
## Class 5 4.21e-02
## Class 6 6.88e-02
## Class 7 4.56e-01
## Class 8 1.65e-01 0.192
## Class 9 6.15e-01
## Class 10 3.88e-01
## Class 11 3.16e-01
## Class 12 1.00e+00 1.000
## Class 13 1.00e+00
## Field 9
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.00e+00
## Class 2 3.13e-16
## Class 3 1.61e-02
## Class 4 6.15e-01
## Class 5 3.46e-02
## Class 6 5.26e-02
## Class 7 1.44e-11
## Class 8 2.09e-01
## Class 9 1.90e-17
## Class 10 8.09e-01
## Class 11 1.00e+00 1.000
## Class 12 7.81e-01 0.703
## Class 13 1.00e+00
## Field 10
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.0000
## Class 2 0.0952
## Class 3 0.1798
## Class 4 0.1741
## Class 5 0.1594
## Class 6 0.1789
## Class 7 0.1208
## Class 8 0.1550
## Class 9 0.2228
## Class 10 0.2602
## Class 11 0.1724
## Class 12 0.3109
## Class 13 1.0000
## Field 11
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.00e+00
## Class 2 6.13e-14
## Class 3 8.84e-07
## Class 4 8.14e-02
## Class 5 2.46e-02
## Class 6 2.13e-02
## Class 7 2.56e-02
## Class 8 3.84e-16
## Class 9 2.44e-01
## Class 10 4.30e-01
## Class 11 3.84e-02
## Class 12 5.86e-01
## Class 13 1.00e+00
## Field 12
## PSRP 1 PSRP 2 PSRP 3 PSRP 4
## Class 1 0.00e+00
## Class 2 2.35e-03
## Class 3 5.57e-02
## Class 4 1.50e-18
## Class 5 2.02e-02
## Class 6 1.67e-02
## Class 7 1.93e-02
## Class 8 4.62e-02
## Class 9 1.85e-02
## Class 10 2.54e-02
## Class 11 5.76e-15
## Class 12 2.26e-01
## Class 13 1.00e+00 1 1 1
## Local Dependence Passing Student Rate
## Field Field Item 1 Field Item 2 Field Item 3 Field Item 4 Parent Class
## 1 1.000 Item01 Item31 Item32 1.000
## 2 2.000 Item11 Item21 Item22 2.000
## 3 2.000 Item11 Item21 Item22 3.000
## 4 3.000 Item23 Item24 4.000
## 5 4.000 Item25 Item26 Item27 5.000
## 6 4.000 Item25 Item26 Item27 7.000
## 7 5.000 Item02 Item03 Item04 2.000
## 8 5.000 Item02 Item03 Item04 4.000
## 9 5.000 Item02 Item03 Item04 6.000
## 10 5.000 Item02 Item03 Item04 8.000
## 11 5.000 Item02 Item03 Item04 10.000
## 12 7.000 Item12 Item13 Item16 Item17 6.000
## 13 8.000 Item28 Item29 6.000
## 14 8.000 Item28 Item29 11.000
## 15 9.000 Item05 Item06 8.000
## 16 9.000 Item05 Item06 9.000
## 17 12.000 Item18 Item19 Item20 Item30 12.000
## Parent CCR 1 Parent CCR 2 Parent CCR 3 Parent CCR 4 Child Class Child CCR 1
## 1 0.000 0.000 0.000 2.000 0.554
## 2 0.005 0.018 0.003 4.000 0.681
## 3 0.034 0.068 0.016 5.000 0.404
## 4 0.221 0.190 5.000 0.189
## 5 0.147 0.050 0.142 6.000 0.613
## 6 0.999 0.991 0.943 11.000 0.941
## 7 0.005 0.040 0.002 3.000 0.073
## 8 0.996 0.998 0.893 7.000 0.145
## 9 0.263 0.334 0.063 9.000 0.294
## 10 0.980 0.958 0.882 12.000 1.000
## 11 0.943 0.800 0.733 12.000 1.000
## 12 0.181 0.146 0.037 0.162 10.000 0.982
## 13 0.009 0.129 8.000 0.165
## 14 0.359 0.273 12.000 1.000
## 15 0.266 0.152 12.000 0.781
## 16 0.000 0.000 11.000 1.000
## 17 0.158 0.178 0.217 0.352 13.000 1.000
## Child CCR 2 Child CCR 3 Child CCR 4
## 1 0.558 0.649
## 2 0.785 0.637
## 3 0.728 0.696
## 4 0.253
## 5 0.436 0.179
## 6 0.925 0.757
## 7 0.330 0.068
## 8 0.265 0.006
## 9 0.298 0.121
## 10 1.000 1.000
## 11 1.000 1.000
## 12 0.883 0.933 0.975
## 13 0.192
## 14 1.000
## 15 0.703
## 16 1.000
## 17 1.000 1.000 1.000
## Marginal Bicluster Reference Matrix
## Class1 Class2 Class3 Class4 Class5 Class6 Class7 Class8 Class9 Class10
## Field1 0 0.587 0.740 0.859 0.875 0.910 0.868 0.889 0.961 0.932
## Field2 0 0.009 0.040 0.701 0.609 0.688 0.832 0.822 1.000 0.984
## Field3 0 0.177 0.219 0.206 0.221 1.000 1.000 1.000 0.986 1.000
## Field4 0 0.013 0.123 0.047 0.113 0.410 0.978 0.954 0.975 0.966
## Field5 0 0.016 0.157 0.963 0.103 0.220 0.138 0.940 0.237 0.825
## Field6 0 0.236 0.275 0.449 0.414 0.302 0.415 0.469 0.560 0.564
## Field7 0 0.073 0.081 0.192 0.160 0.132 0.126 0.179 0.754 0.943
## Field8 0 0.000 0.037 0.039 0.042 0.069 0.456 0.179 0.615 0.388
## Field9 0 0.000 0.016 0.615 0.035 0.053 0.000 0.209 0.000 0.809
## Field10 0 0.095 0.180 0.174 0.159 0.179 0.121 0.155 0.223 0.260
## Field11 0 0.000 0.000 0.081 0.025 0.021 0.026 0.000 0.244 0.430
## Field12 0 0.002 0.056 0.000 0.020 0.017 0.019 0.046 0.019 0.025
## Class11 Class12 Class13
## Field1 0.898 0.975 1
## Field2 1.000 1.000 1
## Field3 0.973 1.000 1
## Field4 0.874 1.000 1
## Field5 0.912 1.000 1
## Field6 0.614 0.764 1
## Field7 0.305 0.786 1
## Field8 0.316 1.000 1
## Field9 1.000 0.742 1
## Field10 0.172 0.311 1
## Field11 0.038 0.586 1
## Field12 0.000 0.226 1
## Class 1 Class 2 Class 3 Class 4 Class 5 Class 6
## Test Reference Profile 0.000 3.900 6.001 12.951 8.853 11.428
## Latent Class Ditribution 2.000 95.000 73.000 37.000 60.000 44.000
## Class Membership Dsitribution 1.987 82.567 86.281 37.258 60.781 43.222
## Class 7 Class 8 Class 9 Class 10 Class 11 Class 12
## Test Reference Profile 14.305 17.148 19.544 23.589 20.343 27.076
## Latent Class Ditribution 43.000 30.000 34.000 18.000 37.000 27.000
## Class Membership Dsitribution 43.062 30.087 34.435 20.063 34.811 25.445
## Class 13
## Test Reference Profile 35
## Latent Class Ditribution 15
## Class Membership Dsitribution 15
##
## Model Fit Indices
## Multigroup Model Saturated Moodel
## model_log_like -5786.942 -5786.942
## bench_log_like -5891.314 0
## null_log_like -9862.114 -9862.114
## model_Chi_sq -208.744 11573.88
## null_Chi_sq 7941.601 19724.23
## model_df 1005 16895
## null_df 1155 17045
## NFI 1 0.4132149
## RFI 1 0.4080052
## IFI 1 1
## TLI 1 1
## CFI 1 1
## RMSEA 0 0
## AIC -2218.744 -22216.12
## CAIC -6486.081 -93954.09
## BIC -6484.132 -93921.32Of course, it also supports various types of plots.
# Show bicluster structure
plot(result.BINET, type = "Array")
# Test Response Profile
plot(result.BINET, type = "TRP")
# Latent Rank Distribution
plot(result.BINET, type = "LRD")
# Rank Membership Profiles for first 9 students
plot(result.BINET, type = "RMP", students = 1:9, nc = 3, nr = 3)
# Field Reference Profiles
plot(result.BINET, type = "FRP", nc = 3, nr = 2)
LDPSR plot shows all Passing Student Rates for all locally dependent classes compared with their respective parents.
# Locally Dependent Passing Student Rates
plot(result.BINET, type = "LDPSR", nc = 3, nr = 2)
Table of Model and Plotting Option Correspondence
| Model/Type | IIC | ICC | TIC | IRP | FRP | TRP | LCD/LRD | CMP/RMP | Array | FieldPIRP | LDPSR |
|---|---|---|---|---|---|---|---|---|---|---|---|
| IRT | ◯ | ◯ | ◯ | ||||||||
| LCA | ◯ | ◯ | ◯ | ◯ | ◯ | ||||||
| LRA | ◯ | ◯ | ◯ | ◯ | ◯ | ||||||
| Biclustering | ◯ | ◯ | ◯ | ◯ | ◯ | ◯ | |||||
| IRM | ◯ | ◯ | ◯ | ||||||||
| LDLRA | ◯ | ◯ | ◯ | ||||||||
| LDB | ◯ | ◯ | ◯ | ◯ | ◯ | ◯ | |||||
| BINET | ◯ | ◯ | ◯ | ◯ | ◯ | ◯ |
Community and Support
We welcome community involvement and feedback to improve
exametrika. Here’s how you can participate and get
support:
Reporting Issues
If you encounter bugs or have suggestions for improvements:
- Open an issue on GitHub Issues
- Provide a minimal reproducible example
- Include your R session information (
sessionInfo())
Discussions and Community
Join our GitHub Discussions:
- Ask questions
- Share your use cases
- Discuss feature requests
- Exchange tips and tricks
- Get updates about package development
Contributing
We appreciate contributions from the community:
- Bug reports and feature requests through Issues
- Usage examples and tips through Discussions
- Code improvements through Pull Requests
Please check our existing Issues and Discussions before posting to avoid duplicates.
Reference
Shojima, Kojiro (2022) Test Data Engineering: Latent Rank Analysis, Biclustering, and Bayesian Network (Behaviormetrics: Quantitative Approaches to Human Behavior, 13),Springer.
Future Updates
Upcoming Features
Polytomous Data Support
- Item Response Theory
- Graded Response Model (GRM)
- Partial Credit Model (PCM)
- Generalized Partial Credit Model (GPCM)
- Latent Structure Analysis
- Polytomous Latent Rank Model
- Polytomous Latent Class Analysis
- Extended Biclustering for polytomous data
Current Development Status
- Binary response models: ✅ Implemented
- Polytomous response models: 🚧 Under development
- CRAN submission: 🚧 In review
Follow our GitHub repository and join the Discussions to stay updated on development progress and provide feedback on desired features.
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